Ground State of a Paramagnetic Impurity in a Metal

A trial wave function first proposed by Kondo as the ground state of the $s-d$ exchange Hamiltonian is analyzed in considerable detail and extended to include potential scattering. A general two-determinant trial wave function in a singlet configuration is shown to lead to a variational energy for antiferromagnetic exchange which is lower than the free-electron energy by an amount ${\epsilon }_S=kD \exp \left[\frac{1}{\mathrm{}(-3\mathrm{}|J\left|{\rho }^0{cos}^2{\delta }^w\right)}\right]$, where $D$ is the conduction-band width, $k\sim 1$, $J$ is the exchange strength, ${\rho }^0$ is the density of states per atom per spin direction, and ${\delta }^w$ is the phase shift due to ordinary scattering at the Fermi surface. It is further shown that the analytic part of the ground-state energy can be obtained by perturbation theory from the nondiagonal (with respect to the variational wave function) part of the Hamiltonian. The remaining terms generated by doing perturbation theory are nonanalytic in $J{\rho }^0$. However, as far as these have been studied, they cause no renormalization of ${\epsilon }_S$ obtained variationally. A triplet ground state has also been studied for ferromagnetic coupling. It is found that everything discussed for the singlet state goes through except that now ${\epsilon }_T=kD \exp \left(\frac{-1\mathrm{}}{J{\rho }^0} {cos}^2{\sigma }^w\right)$. A possible source for the discrepancy between previous theories which indicate no anomalous state for $J>\mathrm{}0\mathrm{}$, as well as a smaller exponential factor ${\epsilon }_S$, is discussed.